Transport: Non-diffusive, flux conservative initial value problems and ...

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64 or multiplying through by ζ we get the quadratic equation which has the solution that ζ 2 + i(2α sin k∆x)ζ − 1 = 0 (5.5.4) ζ = −iαsin k∆x ± � 1 − (α sink∆x) 2 (5.5.5) The norm of ζ now depends on the size of the Courant number α. Since the maximum value of sin k∆x = 1 (which happens for the highest frequency sine wave that can be sampled on the grid) the largest value that α can have before the square root term becomes imaginary is α = 1. Thus for α ≤ 1 �ζ� = 1 (5.5.6) which says that for any value of α ≤ 1. this scheme has no numerical diffusion (which is one of the big advantages of staggered leapfrog). For α > 1, however, the larger root of ζ is �ζ� ∼ � α + � α2 � − 1 which is greater than 1. Therefore the stability requirement is that α ≤ 1 or (5.5.7) ∆t�Vmax� ≤ ∆x (5.5.8) This result is known as the Courant condition and has the physical common-sense interpretation that if you want to maintain accuracy, any particle shouldn’t move more than one grid point per time step. Figure 5.4 shows the behaviour of a gaussian initial condition for α = .9 and α = 1.01 While the staggered-leapfrog scheme is non-diffusive (like our original equation) it can be dispersive at high frequencies and small values of α. If we do a Hirt’s stability analysis on this scheme we get an effective differential equation ∂c ∂c + V ∂t ∂x V � = ∆t 6 2 V 2 − ∆x 2� ∂ 3c ∂x3 = V ∆x2 � α 6 2 � ∂ 3c − 1 ∂x3 (5.5.9) which is dispersive except when α = 1. Unfortunately since α is defined for the maximum velocity in the grid, most regions usually (and should) have α < 1. For well resolved features and reasonable Courant numbers, the dispersion is small. However, high frequency features and excessively small time steps can lead to more noticeable wiggles. Figure 5.5 shows a few examples of dispersion for α = .5 and α = .5 but for a narrower gaussian. In both runs the gaussian travels around the grid 10 times (tmax = 100) (for α = 1 you can’t tell that anything has changed). For many problems a little bit of dispersion will not be important although the wiggles can be annoying looking on a contour plot. If however the small negative values produced by the wiggles will feed back in dangerous ways into your solution you will need a non-dispersive scheme. The most commonly occuring schemes are known as upwind schemes. Before we develop the standard upwind differencing and a iterative improvement on it, however it is useful to develop a slightly different approach to differencing.
Transport equations 65 a concentration b concentration 2.0 1.5 1.0 0.5 2 1 1 0 t=0 1 2 3 0 1 2 3 4 5 6 7 8 9 10 distance t=0 1 2 t=2.53 0 1 2 3 4 5 6 7 8 9 10 distance Figure 5.4: (a) Evolution of gaussian initial condition in using a staggered leapfrog scheme with α = 0.9. (b) α = 1.01 is unstable. 5.5.2 A digression: differencing by the finite volume approach Previously we developed our differencing schemes by considering Taylor series expansions about a point. In this section, we will develop an alternative approach for deriving difference equations that is similar to the way we developed the original conservation equations. This approach will become useful for deriving the upwind and mpdata schemes described below. The control volume approach divides up space into a number of control volumes of width ∆x surrounding each node i.e. and then considers the integral form of the conservation equations d dt � V t=4 � cdV = − cV · dS (5.5.10) s If we now consider that the value of c at the center node of volume j is representa-
Page 1 and 2: Chapter 5 Transport: Non-diffusive,
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